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J. Software Engineering & Applications, 2010, 3: 27-33 doi:10.4236/jsea.2010.31003 Published Online January 2010 (http://www.SciRP.org/journal/jsea) Copyright © 2010 SciRes JSEA 27 Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand Koichi NAKADE, Satoshi TSUBOUCHI, Ibtinen SEDIRI Department of Industrial Engineering and Management, Nagoya Institute of Technology, Japan. Email: nakade@nitech.ac.jp, chd18509@stn.nitech.ac.jp Received September 3rd, 2009; revised October 9th, 2009; accepted October 19th, 2009. ABSTRACT Recently, price contract models between suppliers and retailers, with stochastic demand have been analyzed based on well-known newsvendor problems. In Bernstein and Federgruen [6], they have analyzed a contract model with single supplier and multiples retailers and price dependent demand, where retailers compete on retail prices. Each retailer decides a number of products he procures from the supplier and his retail price to maximize his own profit. This is achieved after giving the wholesale and buy-back prices, which are determined by the supplier as the supplier’s profit is maximized. Bernstein and Federgruen have proved that the retail prices become a unique Nash equilibrium solution under weak conditions on the price dependent distribution of demand. The authors, however, have not mentioned the numerical values and proprieties on these retail prices, the number of products and their individual and overall profits. In this paper, we analyze the model numerically. We first indicate some numerical problems with respect to theorem of Nash equilibrium solutions, which Bernstein and Federgruen proved, and we show their modified results. Then, we compute numerically Nash equilibrium prices, optimal wholesale and buy-back prices for the supplier’s and retailers’ profits, and supply chain optimal retailers’ prices. We also discuss properties on relation between these values and the demand distribution. Keywords: Supply Chain Management, Nash Equilibrium, Stochastic Demand, Competing Retailers 1. Introduction Recently, price contract models between suppliers and retailers with stochastic demand have been analyzed based on well-known newsvendor problems. Cachon [1] has reviewed models with one supplier and one retailer under several types of contracts. In a market, however, many retailers exist and they compete in order to attract the maximum number of consumers. In this context, Yano and Gilbert [2] have been interesting in contracting models in which the demand is stochastic and depends on price. Wang et al. [3] and Petruzzi [4] have studied de- centralized price setting newsvendor problems under multiplicative retail demand functions. Song et al. [5] have analyzed theoretically the optimal prices and the fraction of a total profit under individual optimization to that under supply chain optimization. In Bernstein and Federgruen [6], they have analyzed a contract model with single supplier and multiple retailers and price dependent demand, where retailers compete on retail prices. Each retailer decides a number of products he procures from the supplier and his retail price to maximize his own profit. This is achieved after giving the wholesale and buy-back prices, which are determined by the supplier as the supplier’s profit is maximized. They have proved that the retail prices become a unique Nash equilibrium solution under weak conditions on the price dependent distribution of demand. They, however, have not mentioned the numerical values and properties on these retail prices, the number of products and their individual and overall profits. In this paper, we analyze the model numerically. We first indicate some numerical problems with respect to the theorem of Nash equilibrium solutions, which Bern- stein and Federgruen [6] proved, and we show their modified results. Then we present Nash equilibrium prices, optimal wholesale and buy-back prices for the supplier’s and retailers’ profits, and optimal retail prices under supply chain optimization, analytically and nu- merically. We also discuss the properties on a relation- ship between these values and the demand distribution. In the next section, we present the competing retailers Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand 28 model introduced in [6], and we discuss the sufficient conditions on the existence and the uniqueness of the Nash solution. In Section 3 we investigate the model with exponential and uniform distribution functions and with linear and Logit demand functions. In Section 4, we present numerical results and discuss the behavior of Nash equilibrium solutions and properties of the profits and prices. 2. Competing Retailers’ Model The model of competing retailers for one supplier and retailer introduced in [6], is shown in Figure 1. S N,1 i RiN This model is set under wholesale and buyback pay- ment scheme. The supplier incurs retailer S,1 i R a wholesale price for each product, com- bined with an agreement to buyback unsold inventory at . We assume that the supplier has ample capacity to satisfy any retailer demand and produce products at a constant production cost rate , which includes the transportation cost to retailer . When and are given, each retailer orders his quantity and cho- oses his retail price . A salvage rate is adopted in the supply chain. To avoid trivial setting, the model parameters are chosen as and iN i b i c i R w i i w i w i b ii cv i R i p i y i v ii vb for . 1iN The demand is random and depends on the price vector , with a cumulative dis- tribution function . It is restrained to a multiplicative form i pp, G (x D(p) 12 N p,,p i1 |p,..., p () i Dp d i N ) () i p , where i is a random variable with a cumulative distribution function S R 1 R 2 R i R N c i v i w 1 b 1 w 2 b 2 w i b i w N b N p 1 p 2 p i p N Figure 1. Competing retailers’ model i G(.) and a probability density function i g (.) . In addi- tion i is assumed to be positive only on x and independent of the price vector p. This implies that min [, i xx max ] i ii i x G(x|p) Gd(p) . The demand function depends on the whole price vector. It is supposed that decreases in pi- and increases in pj for all j ≠ i, 1 ≤ I ≤ N, that is () i dp () i dp 0 i i p pd and 0 j i p pd . Let 12 ,,, N y yy y denotes the order vector of the model. The expected profit function for the retailer is given by i R (,)min, ()() iiiiiii pypEyD pbEyD pwy ii , where . It can be rewritten as max 0,a a )()()(),( pDyEbpywpyp iiiiiiii (1) From (1), the retail prices p impact on the profits of all retailers and his order quantity, however, affects only his own profit. In addition the retailer wants to maximize his order quantity. Then, the derivation of the retailer i’s profit function given by Equation (1) on is equal to zero i y 0 i i y ) y ,p( (2) Therefore, the retailer i's optimal corresponding order can be obtained from (1) and (2) by ii ii iii bp wp G)p(d)p(y 1 (3) This result reduces the no-cooperative game in the (p, y)-space to a p-space game. In this space the retailers compete only on prices (reduced retailer game). Then, considering the Equations (1) and (3), we get the retailers profits as a function of p only, as 1 1 ii iiiii ii ii ii ii ii pw (p)d (p)(pw)Gpb pw (pb )EGpb det (| )(()) iiiii pwL fp (4) where is the profit function with a deterministic demand ()p, det (|) ()() iiiii pwpwd p ii yd () () () ii ii ii pw fp pb Copyright © 2010 SciRes JSEA Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand29 is the critical fractile, and i We define and we apply the logarithm (S’), as 1(( )) 11 1 ()()()() / iii Gfp iiiiiii i Lf GffEGfuguduf 1(( )) ˆ() () iii Gfp ii i Lp ugudu to (4), we get for 1iN log og iii i π(p)L (p)loglog l ii (pb)d (p) (5) The supplier profit function is given by ()() () N Miiiiiii wcy bEyDp 1i . Using Equation (3), it can be expressed as 1 M 1 ii ii i dp w cG 1 () . N ii ii ii ii ii ii pw pb pw bvEG pb (6) Differentiating (5) on for i pNi 1 ),p(U p)p(dp ii iii )p(d)p( ~ log ii 1 with 1( i i G f 2 () ) 1 () ()() ii i ii ii iiii wb p Up pb pbLp (7) Bernstein and Federgruen [6] have proved t istence of a Nash solution for the reduced retailer ga the same reeach reta price clos interval o hold: hat the ex- * p me is assured by the following condition (A). (A): For each {1,. .. ,}iNthe function log( ) i dp is increasing in (,) ij pp for all ij. , It is assumed in ference [6] that iler i Rchooses hisi p from aed min max , ii pp . The authors proved the uniqueness of the max Na sh solution in the price space min max , 2, ii i i wbp. This has provided the following conditions (D) and (S) t (D): p 2det 2det 2 log( | iii ji i pw b p log( |)) iii ij pw b pp , (S): x i i i i ix)x(Gdu)u(ug )x(g )x(G x)x( 02 2 ψ, for aanll is the medi of theon under the {1,..., }iN distribution G onditions ma i , where i mx i). However, th n the b (i m e soluti ndabove cy exist oouary of the area i ii pbwp maxmin ,2,max . In this case, it does not satisfy log()0. ip i p the condition (S) is modified to (S’): 2 () ()()0 () ix i xuguduGxx () 2 ii i G xx gx ] Then, the following theorem can ained. S’) hold, th um prices on ψ for all min max [, ii xxx. be obt Theenorem : If conditions (A), (D) and ( there is a unique set of Nash equilibri [,) i i w wtisfy hich salog( )0 ip for all i {1,..., }iN p . Proof: In the same way as in Bnd Federgruen s a unique Nash s[,) i w ernstein a [6], there iolution in * p i which satisfies , because * ii pwfor each {1,..., }iN , () 0 ip when ii p w whereas () 0 ip when ii pw. This ims that plielog( ) i p p 0 i when * pp for all iN{1,. . . ,} . following, the retailerst these um prices, whereas the sthe be- retailers and de k prices t In the sell products a equilibri upplier knows havior of thetermines the wholesale and buybaco maximize his own profit. This system is faces a random de- called “individual optimization”. On the other hand, the problem of determining retail prices and quantities of products to maximize the entire profits of supply chain is called “supply chain optimization” 3. Determination of the Nash Equilibrium As shown in Figure 2, we study a two competing retail- ers’ model. Each retailer {1, 2}i mand (), i Dp where 12 (, )ppp . We assume two types of cumulative distribution functions of demand. We consider first the exponential case and then the uni- form one. 3.1 Exponential Case The cumulative distribution function in the exponential S R 1 c i w 1 1 R 2 b 1 w 2 b 2 p p 2 v i Figure 2. Two competing retailers’ model Copyright © 2010 SciRes JSEA Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand 30 () 1 x i Gxe for all 0x, case is given by where i E is set as rse function of 01y. Wit one without loss () i Gxis given by h of generality. The 1()log(1 ) i Gy y inve for all () () ii ii , we get () ii fp pw pb ) bp bw ()bw(wp bp )p(L iiii ii iiiiii log 1 Then, using (7), we obtain log ii ii ii ii ii i pw U(p ) w pb wb i i i b p wb p 1) Linear demand function The linear demand is given for by j ,, {1,2}jiij () iiiiij ji dpp p with 0,,0 ijii .(8) we obtain the system of equations With this demand function, 11 11 1111122 2 log( )p 2 22 211 log( )()0, ()0. pUp ppp Up p It can be rewritten as 22 22 pp 2222 2222 1 () () , () Up pUp pUp 21 2 2 1 1111111 2 12 11 () () . () Up pUp pUp (9) The optimal order quantities and canbe evaluated to 1 y2 y 11 111 ()yp 1122 11 22 2222211 log , () log pb p pwb pb ypp pw Since 22 . b 1log , iiii i i pwpb wp EG p i i iiiiii bwb pb from (6), the supplier profit function can be expressed as 2 M1 log iii i ii iiii iii ii pb pw dpw c bb wb pb 11 11 11 11 11 22 22 22 22 22 log , log pb pdpbwpw wb pb pdpbw pw wb 2) Logit demand function Now, the problem is studied with a logistic demand function, expressed by 2 () i p i i 1 j p ij j Ck e ke dp for 0and iik,C, . (10) With this demand function we obtain the system of equations 2 12 1 12 112 11 1112 221 22 221 2 log( )()() 0, log( )()()0. p pp p pp pCke Up pCkeke pCke Up pCke ke Then, we have 2 2222 222 1 () () 1log , p C CUpkeUp p 1 12 2 111 11 2 211 () () 1log () p kU p C CUpkeUp pkUp 11 () . The order quantities are given by 1 12 2 12 11 1 11 11 2 22 2 22 21 2 1 2 )log , ()log . pp p pp kep b wb Cke ke kep b yp wb Cke ke The supplier profit function and retailers’ profit func- tio in the same way as for the linear de- ma 3.2 Uniform Case The cumulative distribution function in the uniformase is given by ( p yp ns are obtained nd function. c (1 ) () , 2 i i i Gx a 11, 01,1, 2, ii i xa axaai where 1. i E 1()1Gy ( i The inverse function of is gven by i for . With () i Gx 01y i . The retailers’ profit functions are given by 2 ii a ay ())/( ) ii iii f ppwpb , we get () 1 ii ii iiii ii ii pw pw Lpa a pb pb Copyright © 2010 SciRes JSEA Properties of Nash Equilibrium Retail Prices in Contract Model with a Supplier, Multiple Retailers and Price-Dependent Demand Copyright © 2010 SciRes JSEA 31 r {1, 2}i Then by (7) fo 12 1 () 1 1 ii ii iii iii ii pw pb pwpw aapb 1) Linear demand function With the linear demand given by (8) and , we ii ii ii aa pb wb Up ii )p(U ii obtain 2 222222 ( ) Up 2 1 21 2 2 1111 1111 2 12 11 () ) , ( () () . () Up p pUp Up pUp pUp r quantities are given by The optimal orde 11 111112211 11 22 222221122 22 ()1 2, ()1 2 . pw ypppaa pb pw ypppaapb equal to The supplier profit function is 2 M p 1 2 12 ii iiiiii iii ii ii ii w dpw caapb pw ab pb . The retailers’ profit functions are given by 11 111 1 11 11 2 11 11 1 11 22 2 22 22 2 22 22 2 22 12 , . pwa apb pdp pw ap bpb pw pb pdp pw ap bpb 2) Logit demand function With the Logit function given by (10), we obtain and as 22 2 12pw aa pw 1 p2 p 2 1 2222 222 1 122 1111111 2 211 1log , 1log . λp λCCU(p)keU λp (p) pλkλU(p) λCCU(p )keU(p ) pλkλU(p) The optimal order quantities are given by 1 12 2 12 11 11 11 11 2 22 22 22 21 2 ()1 2, ()1 2. p pp p pp kepw ypaa pb Cke ke kep w ypaa pb Cke ke 1 1 2 2 The supplier profit function and retailers’ profit func- tions are obtained in the same way as for the linear de- mand function. 3.zation W anordantities to maximize the overall profit of the supply chain, the wholesale and buyback prices are meaningless because they are payments between the supplier and the retailers. As the whole of the supply chain is equivalent to a single retailer with who price and buyback 3 Supply Chain Optimi hen the supplier and the retailers determine the prices d the er qu lesale 12 (, )cc 12 (, ) , and by using (3), the optimal order quantity (the amount of products) is given by 1 () () Iii iii pc yp dpG ii p . By u sing (4), the overall expected profit of the supply chain is 2 1 ()() () Iii iii i iii pc ppcdpL pv , (11) wh ere, the retail prices 12 (, )pp are given. The optimal retail prices 12 (,) I I pp in the integrated supply chain maximize the profit function given by (11). ote th 4. Numerical Examples 4.1 Geometric Analysis of the Nash Solution The system of equations on 12 (, )pp that solves the profit functions for the two retailers is obtained in Section 3. In the case of exponential demand and linear functions, we dene right hand sides of two equations in (9) by 22 () f p and 11 () f p, respectively. Then the equations (9e ) becom12 ()pf2 p and Note that in s sa 21 ().pfp tisfied 1 other caseby (, )pp form 122 ()pfp the equations 12 and 211 ()pfp similarly. Geom he etrically, Nash to analyze the ution, we behavior ot the f of th ncti e system around t ons () ii sol plu f p fo le solutions fo solution r 1 pand 2 p r the equa-in Figure 3. There are multip e Nashtions, but there is a uniqu12 (,)pp with (1,2) ii pwi, which has been proved in the theorem of Section 2. 32 P 1 P 2 w 2 w 1 12 (,)pp Figure 3. Nash solution and system of equations Given wholesale and buyback prices, we derive these Nash retail prices, and profits of the supplier and two retailers. We compute them for all combinations of wholesale and buyback prices, which are integers and satisfy and i U iii cww ii vbw , where is set as the upper bound for the optimal wholesale price for the supplier, and derive optimal wholesale and buyback prices for the supplier. We also compute the overall prof- its and retail prices under the supply chain optimization, and compare them with the ones under individual i- mization. .2 Numerical Results U i w opt 4 In numerical examples we set parameters as shown in the following: 12 (, ) (0,0) , 12 ( ,))(100,100 ，12 (, )(1,1) ， 12 21 (,)(0.3,0.3) (linear function ) , 0.03 ，12 ( ,)(0.005,0.005)CC ， 12 (, ) (1,1)kk , (Logit function). The program is coded by C and the computations are mpiler on PC. In Table 1, we d Logit functions, wher- cost parameter settings are considered: etric) and done by using Fujitsu C co assume exponential demand an eas in Table 2 the linear function is assumed. In these tables two 12 ( ,)(30,30)cc (symm12 (,)(30, 20)cc (anti-symmetiric). The values in tables are the optimal pfor supplier, the profit for each retaim of supplier’s and retailers’ profits), optimal whole-sale and buyback prices for the supplier, Nashilibrium retail prices and orderes in paranthesis () are th function i rofit ler; entire expected profit (su equ quantities. The valu e total profit, optimal retail prices and order quanti- ties for retailers under the supply chain optimization. In the cases of Tables 1 and 2, optimal whole sale prices and buybacks determined by the supplier give more profits to the supplier than retailers. In the symme- Table 1. Exponential demand and logit 1 2 1 2 Ci 30 30 30 20 () M p 32.195 35.792 (,) iii py 10.227 10.227 7 3.843 Entire expected profits 52.649 (62.430) 58.552 (7 8.91 1 0.153) i w 98 98 100 88 i b 47 47 47 47 175.420 (172.428) 175.420 (172.428) 175.376 (182.095) 168.444 (161.07) 0.311 0.311 0.276 0.418 i p i y (0.606) (0.606) (0.444) (0.965) ential demand and linear function Table 2. Expon i a 0.1 0.3 0.5 0.7 () M p 2531.42 2352.36 2176.38 2003.38 (,) iii py 513.03 481.51 450.56 414.12 Entire expected profits 3557.49 (4303.71) 3315.38 (3999.12) 3077.51 (3700.00) 2832.62 (3407.00) 12 ()ww87 87 87 87 12 ()bb75 75 75 74 1 () 2 p p110.31 110.97 111.69 112.55 ) (87.08) (88.46) (89.96) (91.56 2 (4) 2 (4) 2 (4) 2 (4) 12 ()yy3.51 0.26 4.55 1.75 5.59 3.20 6.05 4.57 tric cost casesptimil prf two retailers bec same. red to supply ptimiza- tioail pricgher and the s of or- ders are smaller in the individual optimal case. It is be- cause uer then optiation mamountf de- mand a satisf decsing re prices in- creasing ordeiess idivti- mal case ther o ofit, which leads toer wle prnd asult retail prices bighemmost ase, the optimal wholesale price to the retailer with the smalther retailer, which leads to mrofits he fo- tailer. The reaso that thailer wmall w price setshe il d mntr- der, which imat mouur in td the supplier can sell more products to cus- tompa wit he demepethri ly, and the wholespricestail prind th quantities changore. In both cases entirepected profits in indi- vidual optimalis a0 toof er supply chain on tn cof , the oal retaices o ome the n, the ret Compa chain o es are hiquantitie nd chaimizore s o reied byreatail and r quantit, wherean the inidual op suppliewants to btain itswn pro highholesaices as a re ecome hher. In t anti-syetric c c ler production cost is smaller than that to ano ore p e ret for t ith s rmer re holesale n is tless retaprice anore quaities of o plies thore amnts of demand occ otal an ers. In and d rticular,ith Logdemand function t nds on e retail pces more intensive ale , reces ae order e m the exthe cases bout 8 85 % that und optimizatin. Whehe chaionsists Copyright © 2010 SciRes JSEA Copyright © 2010 SciRes JSEA 33 Table 3. Uniform deand unct i mand linear fion 1 2 1 2 Ci30 30 30 20 () M p 1200.548 1473.307 (,) iii p y 242.306 242.306 228.119 380.888 Entire expected profits 1685.160 (2041.22) 2082.314 (2515.01) i w 89 89 89 82 i b 77 77 77 73 i p 116.154 (96.902) 116.154 (96.902) 115.532 (97.788) 112.445 (90.259) i y 22.105 (37.717) 22.105 (37.717) 21.233 (34.608) 32.826 (58.887) one supplier and one retailer, it is shown in Song et al. (2008) that the fraction is 3/4(in linear case) or 2 /0.736e (in Logit case). The competition among retailers makes retail prices lower, which makes the frac- tion higher. In Table 3, the uniform distribution of demand is assumed with the symmetric production costs ((12 , )(30,30)cc ), and the i a, which corresponds to the width of the uniform distribution, is changed from 0.1 to 0.7. It implies that large i a means the high variance of dems are higher, d profitsf the sulier andtailers drease. This is because when the vace inces, theantity of order mt be into apply tation of demand, whereas thelso incd to obtain proitsle Wchanges the optimal wholesale prices an buybes for tr are almoe. Note that even it is cared w result the exponential case shown in Fige 2, whi h has mvariance these unif disttions, difference on the is very small. ohand buyback pricese tnce of the demand d tail prices. In numerical ex- am rant-in-Aid for Scientific t with revenue sharing,” Man- hain modeling implications and in- sights,” Workusiness, University of Illinois, Urb4. 5. Concluding Remarks In this paper, we first show the sufficient condition that unique Nash equilibrium retail prices exist and they are greater than wholesale prices. We then give the equations whose solutions are those re ples we compute these equilibrium prices, optimal wholesale and buy-back prices for the supplier and sup- ply chain optimal retailers’ prices, and discuss properties on these values. In future research, a two-supplier prob- lem and other types of problems will be modeled and the properties will be discussed analytically and numerically. 6. Acknowledgments This work was supported by G Research (C) 19510145. REFERENCES [1] G. P. Cachon, “Supply chain coordination with contracts, in supply chain management, design, coordination and operation,” Elsevier, Vol. 11, Amsterdam, pp. 229–339. [2] C. A. Yano and S. M. Gilbert, “Coordinated pricing and production/procurement decisions: A review,” A. Chak- ravarty, J. Eliashberg, eds. “Managing business interfaces: marketing, engineering and manufacturing perspectives,” Kluwer Academic Publishers, Boston, MA, 2003. ., “Channel performance un-[3] Y. Wang, J. Li, and Z. Shen der consignment contrac and. As the variance increases, retail price an opp reec rianreas qu uscreased he fluctu retail of retai price m rs. ust be arease f hen i a ack pric agement Sci. Vol. 50, No. 1, pp. 34–47, January 2004. [4] N. C. Petruzzi, “Newsvendor pricing, purchasing and consignment: Supply c ing Paper, College of B ana-Champaign, IL, 200 d [5] Y. Song, S. Ray, and S. Li, “Structural properties of buy- back contracts for price-setting newsvendors,” MSOM, Vol. 10, pp. 1–18, January 2007. he suppliest the sam ifompiths in ur ribu c the ore than [6] F. Bernstein and A. Federgruen, “Decentralized supply chains with competing retailers under demand uncer- tainty,” Management Science, Vol. 51, pp. 18–29, Janu- ary 2005. orm se prices It means that the ptimal wolesale for the supplier arrobust inhe varia istribution. |